<p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_817_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="81" /> </InlineMediaObject> <EquationSource Format="TEX">$\psi \in H(\mathbb {B}_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ψ</mi> <mo>∈</mo> <mi>H</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">B</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be the space of all holomorphic functions on the unit ball <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_817_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {B}_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">B</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_817_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {C}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_817_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="184" /> </InlineMediaObject> <EquationSource Format="TEX">$\varphi = (\varphi _1, \ldots , \varphi _n) \in S(\mathbb {B}_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>φ</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <mi>S</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">B</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> the set of holomorphic self-maps of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_817_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {B}_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">B</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>. Let <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_817_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="257" /> </InlineMediaObject> <EquationSource Format="TEX">$W_{\psi , \varphi }:\mathscr {B}_\nu \, (\text {or }\mathscr {B}_{\nu ,0}) \rightarrow \mathscr {B}_\mu \, (\text {or } \mathscr {B}_{\mu ,0})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>W</mi> <mrow> <mi>ψ</mi> <mo>,</mo> <mi>φ</mi> </mrow> </msub> <mo>:</mo> <msub> <mi mathvariant="script">B</mi> <mi>ν</mi> </msub> <mspace width="0.166667em" /> <mrow> <mo stretchy="false">(</mo> <mtext>or</mtext> <mspace width="0.333333em" /> <msub> <mi mathvariant="script">B</mi> <mrow> <mi>ν</mi> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <msub> <mi mathvariant="script">B</mi> <mi>μ</mi> </msub> <mspace width="0.166667em" /> <mrow> <mo stretchy="false">(</mo> <mtext>or</mtext> <mspace width="0.333333em" /> <msub> <mi mathvariant="script">B</mi> <mrow> <mi>μ</mi> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_817_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="102" /> </InlineMediaObject> <EquationSource Format="TEX">$f\mapsto \psi \hspace{1.111pt}{\cdot }\hspace{1.111pt}(f \hspace{0.55542pt}{\circ }\hspace{1.111pt}\varphi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>↦</mo> <mi>ψ</mi> <mspace width="1.111pt" /> <mo>·</mo> <mspace width="1.111pt" /> <mo stretchy="false">(</mo> <mi>f</mi> <mspace width="0.55542pt" /> <mo>∘</mo> <mspace width="1.111pt" /> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, be weighted composition operators between (little) Bloch-type spaces where <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_817_Article_IEq8.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="30" /> </InlineMediaObject> <EquationSource Format="TEX">$\nu , \mu $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ν</mi> <mo>,</mo> <mi>μ</mi> </mrow> </math></EquationSource> </InlineEquation> are normal weights on <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_817_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {B}_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">B</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>. We characterize the boundedness, compactness and give the asymptotic estimates of the norms of <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_817_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">$ W_{\psi ,\varphi } $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>W</mi> <mrow> <mi>ψ</mi> <mo>,</mo> <mi>φ</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> in terms of function theoretic properties of the symbol <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_817_Article_IEq11.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ψ</mi> </math></EquationSource> </InlineEquation> and of the modulus <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_817_Article_IEq12.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">$|\varphi _k|$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi>φ</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">|</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of a component function <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_817_Article_IEq13.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$\varphi _k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>φ</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> of the holomorphic symbol <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_817_Article_IEq14.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation>.</p>

Some new characterizations of the boundedness and compactness of weighted composition operators between Bloch-type spaces

期刊论文
发表日期：

2025年4月11日

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Lien Vuong Lam,
Nguyen Van Dai,
Thai Thuan Quang

摘要

Let $\psi \in H(\mathbb {B}_n)$ $ψ \in H (B_{n})$ be the space of all holomorphic functions on the unit ball $\mathbb {B}_n$ $B_{n}$ of $\mathbb {C}^n$ $C^{n}$ and $\varphi = (\varphi _1, \ldots , \varphi _n) \in S(\mathbb {B}_n)$ $φ = (φ_{1}, \dots, φ_{n}) \in S (B_{n})$ the set of holomorphic self-maps of $\mathbb {B}_n$ $B_{n}$ . Let $W_{\psi , \varphi }:\mathscr {B}_\nu \, (\text {or }\mathscr {B}_{\nu ,0}) \rightarrow \mathscr {B}_\mu \, (\text {or } \mathscr {B}_{\mu ,0})$ $W_{ψ, φ} : B_{ν} (or B_{ν, 0}) \to B_{μ} (or B_{μ, 0})$ , $f\mapsto \psi \hspace{1.111pt}{\cdot }\hspace{1.111pt}(f \hspace{0.55542pt}{\circ }\hspace{1.111pt}\varphi )$ $f \mapsto ψ \cdot (f \circ φ)$ , be weighted composition operators between (little) Bloch-type spaces where $\nu , \mu $ $ν, μ$ are normal weights on $\mathbb {B}_n$ $B_{n}$ . We characterize the boundedness, compactness and give the asymptotic estimates of the norms of $ W_{\psi ,\varphi } $ $W_{ψ, φ}$ in terms of function theoretic properties of the symbol $\psi $ $ψ$ and of the modulus $|\varphi _k|$ $| φ_{k} |$ of a component function $\varphi _k$ $φ_{k}$ of the holomorphic symbol $\varphi $ $φ$ .